DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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Arithmetic functions on the positive integers

Definition

An arithmetic function is a function

$f : Z_{> 0} \to C,$

where $Z_{> 0}$ is the positive part of the divisibility poset of The divisibility poset of positive integers and $C$ is the field of $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ .

The domain is exactly the positive integers. In particular, no arithmetic function value at $0$ is part of the present convention, and every later divisor sum on this page ranges over positive divisors only.

Remarks

The codomain is fixed to $C$ so that pointwise addition, products, and finite divisor sums all live in one ambient field.

Depends on

Used by

Completely multiplicative arithmetic functions Definition
Dirichlet convolution of arithmetic functions Definition
Liouville's function Definition
Multiplicative arithmetic functions Definition
The Dirichlet-convolution identity and the constant-one function Definition
The divisor-counting function τ Definition
The power functions idₖ and the divisor-power-sum functions σₖ Definition
The von Mangoldt function Definition

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 2.9 (standard reference, not scraped)
Kiran S. Kedlaya, An Introduction to Analytic Number Theory, Chapter 3 (standard reference, not scraped)